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1 (* ID: $Id$ |
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2 Authors: Klaus Aehlig, Tobias Nipkow |
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3 *) |
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4 |
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5 header {* Alternativ implementation of "normalization by evaluation" for HOL, including test examples *} |
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6 |
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7 theory Nbe |
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8 imports Main |
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9 uses |
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10 "~~/src/Tools/Nbe/nbe_eval.ML" |
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11 "~~/src/Tools/Nbe/nbe_package.ML" |
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12 begin |
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13 |
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14 ML {* reset Toplevel.debug *} |
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15 |
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16 setup Nbe_Package.setup |
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17 |
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18 method_setup normalization = {* |
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19 Method.no_args (Method.SIMPLE_METHOD' |
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20 (CONVERSION (ObjectLogic.judgment_conv Nbe_Package.normalization_conv) |
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21 THEN' resolve_tac [TrueI, refl])) |
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22 *} "solve goal by normalization" |
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23 |
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24 |
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25 text {* lazy @{const If} *} |
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26 |
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27 definition |
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28 if_delayed :: "bool \<Rightarrow> (bool \<Rightarrow> 'a) \<Rightarrow> (bool \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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29 [code func del]: "if_delayed b f g = (if b then f True else g False)" |
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30 |
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31 lemma [code func]: |
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32 shows "if_delayed True f g = f True" |
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33 and "if_delayed False f g = g False" |
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34 unfolding if_delayed_def by simp_all |
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35 |
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36 lemma [normal pre, symmetric, normal post]: |
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37 "(if b then x else y) = if_delayed b (\<lambda>_. x) (\<lambda>_. y)" |
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38 unfolding if_delayed_def .. |
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39 |
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40 hide (open) const if_delayed |
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41 |
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42 lemma "True" by normalization |
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43 lemma "x = x" by normalization |
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44 lemma "True \<or> False" |
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45 by normalization |
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46 lemma "True \<or> p" by normalization |
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47 lemma "p \<longrightarrow> True" |
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48 by normalization |
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49 declare disj_assoc [code func] |
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50 lemma "((P | Q) | R) = (P | (Q | R))" by normalization |
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51 declare disj_assoc [code func del] |
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52 lemma "0 + (n::nat) = n" by normalization |
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53 lemma "0 + Suc n = Suc n" by normalization |
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54 lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization |
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55 lemma "~((0::nat) < (0::nat))" by normalization |
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56 |
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57 datatype n = Z | S n |
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58 consts |
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59 add :: "n \<Rightarrow> n \<Rightarrow> n" |
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60 add2 :: "n \<Rightarrow> n \<Rightarrow> n" |
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61 mul :: "n \<Rightarrow> n \<Rightarrow> n" |
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62 mul2 :: "n \<Rightarrow> n \<Rightarrow> n" |
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63 exp :: "n \<Rightarrow> n \<Rightarrow> n" |
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64 primrec |
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65 "add Z = id" |
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66 "add (S m) = S o add m" |
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67 primrec |
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68 "add2 Z n = n" |
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69 "add2 (S m) n = S(add2 m n)" |
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70 |
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71 lemma [code]: "add2 (add2 n m) k = add2 n (add2 m k)" |
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72 by(induct n) auto |
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73 lemma [code]: "add2 n (S m) = S (add2 n m)" |
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74 by(induct n) auto |
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75 lemma [code]: "add2 n Z = n" |
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76 by(induct n) auto |
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77 |
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78 lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization |
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79 lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization |
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80 lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization |
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81 |
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82 primrec |
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83 "mul Z = (%n. Z)" |
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84 "mul (S m) = (%n. add (mul m n) n)" |
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85 primrec |
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86 "mul2 Z n = Z" |
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87 "mul2 (S m) n = add2 n (mul2 m n)" |
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88 primrec |
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89 "exp m Z = S Z" |
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90 "exp m (S n) = mul (exp m n) m" |
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91 |
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92 lemma "mul2 (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization |
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93 lemma "mul (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization |
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94 lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization |
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95 |
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96 lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization |
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97 lemma "split (%x y. x) (a, b) = a" by normalization |
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98 lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization |
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99 |
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100 lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization |
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101 |
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102 lemma "[] @ [] = []" by normalization |
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103 lemma "[] @ xs = xs" by normalization |
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104 normal_form "[a, b, c] @ xs = a # b # c # xs" |
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105 normal_form "map f [x,y,z::'x] = [f x, f y, f z]" |
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106 normal_form "map (%f. f True) [id, g, Not] = [True, g True, False]" |
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107 normal_form "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" |
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108 normal_form "rev [a, b, c] = [c, b, a]" |
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109 normal_form "rev (a#b#cs) = rev cs @ [b, a]" |
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110 normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])" |
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111 normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))" |
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112 normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])" |
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113 normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()]" |
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114 normal_form "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False" |
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115 normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs" |
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116 normal_form "let x = y::'x in [x,x]" |
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117 normal_form "Let y (%x. [x,x])" |
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118 normal_form "case n of Z \<Rightarrow> True | S x \<Rightarrow> False" |
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119 normal_form "(%(x,y). add x y) (S z,S z)" |
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120 normal_form "filter (%x. x) ([True,False,x]@xs)" |
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121 normal_form "filter Not ([True,False,x]@xs)" |
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122 |
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123 normal_form "[x,y,z] @ [a,b,c] = [x, y, z, a, b ,c]" |
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124 normal_form "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" |
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125 normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" |
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126 |
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127 lemma "last [a, b, c] = c" |
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128 by normalization |
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129 lemma "last ([a, b, c] @ xs) = (if null xs then c else last xs)" |
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130 by normalization |
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131 |
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132 lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization |
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133 lemma "(-4::int) * 2 = -8" by normalization |
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134 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization |
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135 lemma "(2::int) + 3 = 5" by normalization |
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136 lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization |
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137 lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization |
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138 lemma "(2::int) < 3" by normalization |
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139 lemma "(2::int) <= 3" by normalization |
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140 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization |
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141 lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization |
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142 lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization |
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143 lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization |
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144 lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization |
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145 |
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146 normal_form "Suc 0 \<in> set ms" |
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147 |
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148 normal_form "f" |
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149 normal_form "f x" |
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150 normal_form "(f o g) x" |
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151 normal_form "(f o id) x" |
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152 normal_form "\<lambda>x. x" |
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153 |
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154 (* Church numerals: *) |
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155 |
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156 normal_form "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |
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157 normal_form "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |
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158 normal_form "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |